在高度平衡树中使用后退指针是什么意思？

Question

我必须根据高度平衡的树来编写带有背景的高度平衡树代码 代码如下。我必须以没有堆栈的方式修改下面的代码 在重新平衡期间再次使用以跟踪路径;而是每个非根 节点应该有一个额外的字段，指向节点的上邻居。 这些字段需要正确设置，特别是在旋转中，以及每当执行时 在叶级插入或删除。

#include <stdio.h>
#include <stdlib.h>

#define BLOCKSIZE 256

typedef int object_t;
typedef int key_t;
typedef struct tr_n_t { key_t        key;
                    struct tr_n_t  *left;
                    struct tr_n_t *right;
                    int           height; 
                      } tree_node_t;



tree_node_t *currentblock = NULL;
int    size_left;
tree_node_t *free_list = NULL;

tree_node_t *get_node()
{ tree_node_t *tmp;
  if( free_list != NULL )
  {  tmp = free_list;
     free_list = free_list -> left;
  }
  else
  {  if( currentblock == NULL || size_left == 0)
     {  currentblock = 
                (tree_node_t *) malloc( BLOCKSIZE * sizeof(tree_node_t) );
        size_left = BLOCKSIZE;
     }
     tmp = currentblock++;
     size_left -= 1;
  }
  return( tmp );
}


void return_node(tree_node_t *node)
{  node->left = free_list;
   free_list = node;
}


tree_node_t *create_tree(void)
{  tree_node_t *tmp_node;
   tmp_node = get_node();
   tmp_node->left = NULL;
   return( tmp_node );
}

void left_rotation(tree_node_t *n)
{  tree_node_t *tmp_node;
   key_t        tmp_key;
   tmp_node = n->left; 
   tmp_key  = n->key;
   n->left  = n->right;        
   n->key   = n->right->key;
   n->right = n->left->right;  
   n->left->right = n->left->left;
   n->left->left  = tmp_node;
   n->left->key   = tmp_key;
}

void right_rotation(tree_node_t *n)
{  tree_node_t *tmp_node;
   key_t        tmp_key;
   tmp_node = n->right; 
   tmp_key  = n->key;
   n->right = n->left;        
   n->key   = n->left->key;
   n->left  = n->right->left;  
   n->right->left = n->right->right;
   n->right->right  = tmp_node;
   n->right->key   = tmp_key;
}

object_t *find(tree_node_t *tree, key_t query_key)
{  tree_node_t *tmp_node;
   if( tree->left == NULL )
     return(NULL);
   else
   {  tmp_node = tree;
      while( tmp_node->right != NULL )
      {   if( query_key < tmp_node->key )
               tmp_node = tmp_node->left;
          else
               tmp_node = tmp_node->right;
      }
      if( tmp_node->key == query_key )
         return( (object_t *) tmp_node->left );
      else
         return( NULL );
   }
}


int insert(tree_node_t *tree, key_t new_key, object_t *new_object)
{  tree_node_t *tmp_node;
   int finished;
   if( tree->left == NULL )
   {  tree->left = (tree_node_t *) new_object;
      tree->key  = new_key;
      tree->height = 0;
      tree->right  = NULL; 
   }
   else
     {  tree_node_t * path_stack[100]; int  path_st_p = 0;
      tmp_node = tree; 
      while( tmp_node->right != NULL )
      {   path_stack[path_st_p++] = tmp_node;
          if( new_key < tmp_node->key )
               tmp_node = tmp_node->left;
          else
               tmp_node = tmp_node->right;
      }
      /* found the candidate leaf. Test whether key distinct */
      if( tmp_node->key == new_key )
         return( -1 );
      /* key is distinct, now perform the insert */ 
      {  tree_node_t *old_leaf, *new_leaf;
         old_leaf = get_node();
         old_leaf->left = tmp_node->left; 
         old_leaf->key = tmp_node->key;
         old_leaf->right  = NULL;
         old_leaf->height = 0;
         new_leaf = get_node();
         new_leaf->left = (tree_node_t *) new_object; 
         new_leaf->key = new_key;
         new_leaf->right  = NULL;
         new_leaf->height = 0; 
         if( tmp_node->key < new_key )
         {   tmp_node->left  = old_leaf;
             tmp_node->right = new_leaf;
             tmp_node->key = new_key;
         } 
         else
         {   tmp_node->left  = new_leaf;
             tmp_node->right = old_leaf;
         } 
         tmp_node->height = 1;
      }
      /* rebalance */
      finished = 0;
      while( path_st_p > 0 && !finished )
      {  int tmp_height, old_height;
         tmp_node = path_stack[--path_st_p];
         old_height= tmp_node->height;
         if( tmp_node->left->height - 
                                 tmp_node->right->height == 2 )
         {  if( tmp_node->left->left->height - 
                                 tmp_node->right->height == 1 )
            {  right_rotation( tmp_node );
               tmp_node->right->height = 
                            tmp_node->right->left->height + 1;
               tmp_node->height = tmp_node->right->height + 1;
            }
            else
            {  left_rotation( tmp_node->left );
               right_rotation( tmp_node );
               tmp_height = tmp_node->left->left->height; 
               tmp_node->left->height  = tmp_height + 1; 
               tmp_node->right->height = tmp_height + 1; 
               tmp_node->height = tmp_height + 2; 
            }
         }
         else if ( tmp_node->left->height - 
                                tmp_node->right->height == -2 )
         {  if( tmp_node->right->right->height - 
                                  tmp_node->left->height == 1 )
            {  left_rotation( tmp_node );
               tmp_node->left->height = 
                           tmp_node->left->right->height + 1;
               tmp_node->height = tmp_node->left->height + 1;
            }
            else
            {  right_rotation( tmp_node->right );
               left_rotation( tmp_node );
               tmp_height = tmp_node->right->right->height; 
               tmp_node->left->height  = tmp_height + 1; 
               tmp_node->right->height = tmp_height + 1; 
               tmp_node->height = tmp_height + 2; 
            }
         }
         else /* update height even if there was no rotation */ 
         {  if( tmp_node->left->height > tmp_node->right->height )
               tmp_node->height = tmp_node->left->height + 1;
            else
               tmp_node->height = tmp_node->right->height + 1;
         }
         if( tmp_node->height == old_height )
            finished = 1;
      }

   }
   return( 0 );
}



object_t *delete(tree_node_t *tree, key_t delete_key)
{  tree_node_t *tmp_node, *upper_node, *other_node;
   object_t *deleted_object; int finished;
   if( tree->left == NULL )
      return( NULL );
   else if( tree->right == NULL )
   {  if(  tree->key == delete_key )
      {  deleted_object = (object_t *) tree->left;
         tree->left = NULL;
         return( deleted_object );
      }
      else
         return( NULL );
   }
   else
   {  tree_node_t * path_stack[100]; int path_st_p = 0;
      tmp_node = tree;
      while( tmp_node->right != NULL )
      {   path_stack[path_st_p++] = tmp_node;  
          upper_node = tmp_node;
          if( delete_key < tmp_node->key )
          {  tmp_node   = upper_node->left; 
             other_node = upper_node->right;
          } 
          else
          {  tmp_node   = upper_node->right; 
             other_node = upper_node->left;
          } 
      }
      if( tmp_node->key != delete_key )
         deleted_object = NULL;
      else
      {  upper_node->key   = other_node->key;
         upper_node->left  = other_node->left;
         upper_node->right = other_node->right;
         upper_node->height = other_node->height;
         deleted_object = (object_t *) tmp_node->left;
         return_node( tmp_node );
         return_node( other_node );

      }
      /*start rebalance*/  
      finished = 0; path_st_p -= 1;
      while( path_st_p > 0 && !finished )
      {  int tmp_height, old_height;
         tmp_node = path_stack[--path_st_p];
         old_height= tmp_node->height;
         if( tmp_node->left->height - 
                                 tmp_node->right->height == 2 )
         {  if( tmp_node->left->left->height - 
                                 tmp_node->right->height == 1 )
        {  right_rotation( tmp_node ); 
               tmp_node->right->height = 
                            tmp_node->right->left->height + 1;
               tmp_node->height = tmp_node->right->height + 1;
            }
            else
        {  left_rotation( tmp_node->left ); 
               right_rotation( tmp_node );
               tmp_height = tmp_node->left->left->height; 
               tmp_node->left->height  = tmp_height + 1; 
               tmp_node->right->height = tmp_height + 1; 
               tmp_node->height = tmp_height + 2; 
            }
         }
         else if ( tmp_node->left->height - 
                                tmp_node->right->height == -2 )
         {  if( tmp_node->right->right->height - 
                                  tmp_node->left->height == 1 )
        {  left_rotation( tmp_node ); 
               tmp_node->left->height = 
                           tmp_node->left->right->height + 1;
               tmp_node->height = tmp_node->left->height + 1;
            }
            else
            {  right_rotation( tmp_node->right );
               left_rotation( tmp_node );
               tmp_height = tmp_node->right->right->height; 
               tmp_node->left->height  = tmp_height + 1; 
               tmp_node->right->height = tmp_height + 1; 
               tmp_node->height = tmp_height + 2; 
            }
         }
         else /* update height even if there was no rotation */ 
         {  if( tmp_node->left->height > tmp_node->right->height )
               tmp_node->height = tmp_node->left->height + 1;
            else
               tmp_node->height = tmp_node->right->height + 1;
         }
         if( tmp_node->height == old_height )
        finished = 1;
      }
      /*end rebalance*/
      return( deleted_object );
   }
}


void check_tree( tree_node_t *tr, int depth, int lower, int upper )
{  if( tr->left == NULL )
   {  printf("Tree Empty\n"); return; }
   if( tr->key < lower || tr->key >= upper )
         printf("Wrong Key Order \n");
   if( tr->right == NULL )
   {  if( *( (int *) tr->left) == 10*tr->key + 2 )
         printf("%d(%d)  ", tr->key, depth );
      else
         printf("Wrong Object \n");
   }
   else
   {  check_tree(tr->left, depth+1, lower, tr->key ); 
      check_tree(tr->right, depth+1, tr->key, upper ); 
   }
}

int main()
{  tree_node_t *searchtree;
   char nextop;
   searchtree = create_tree();
   printf("Made Tree: Height-Balanced Tree\n");
   while( (nextop = getchar())!= 'q' )
   { if( nextop == 'i' )
     { int inskey,  *insobj, success;
       insobj = (int *) malloc(sizeof(int));
       scanf(" %d", &inskey);
       *insobj = 10*inskey+2;
       success = insert( searchtree, inskey, insobj );
       if ( success == 0 )
         printf("  insert successful, key = %d, object value = %d, \
                  height is %d\n",
              inskey, *insobj, searchtree->height );
       else
           printf("  insert failed, success = %d\n", success);
     }  
     if( nextop == 'f' )
     { int findkey, *findobj;
       scanf(" %d", &findkey);
       findobj = find( searchtree, findkey);
       if( findobj == NULL )
         printf("  find failed, for key %d\n", findkey);
       else
         printf("  find successful, found object %d\n", *findobj);
     }
     if( nextop == 'd' )
     { int delkey, *delobj;
       scanf(" %d", &delkey);
       delobj = delete( searchtree, delkey);
       if( delobj == NULL )
         printf("  delete failed for key %d\n", delkey);
       else
         printf("  delete successful, deleted object %d, height is now %d\n", 
             *delobj, searchtree->height);
     }
     if( nextop == '?' )
     {  printf("  Checking tree\n"); 
        check_tree(searchtree,0,-1000,1000);
        printf("\n");
        if( searchtree->left != NULL )
      printf("key in root is %d, height of tree is %d\n", 
         searchtree->key, searchtree->height );
        printf("  Finished Checking tree\n"); 
     }
   }
   return(0);
}

“使用backpointers”和“不再使用堆栈”是什么意思？我是否必须修改/* start rebalancing */部分以及函数rotation和insert？我有点理解高度平衡的树是如何工作的，但我并没有真正得到我为这项任务所做的事情。

Answer 1

在您的起始树结构中，每个节点都有指向其左右子节点（如果有）的指针，但不指向其父节点。如果您需要在这样的树上执行操作，该树需要知道从树的根到某个感兴趣的节点的部分或全部路径，那么您需要通过遍历来构造该路径树和记录路径 - 例如，在堆栈数据结构中。您无法从末端节点向后工作。

您可以在发布的代码中看到此类行为。例如，在函数insert()中，您有......

         tree_node_t * path_stack[100]; int  path_st_p = 0;

......以后......

             path_stack[path_st_p++] = tmp_node;

......等等。

另一方面，如果每个节点都有一个指向其父节点的指针，则不需要跟踪树中的路径。相反，您可以从任何节点开始，并根据需要向后向上移动树，因为这样做所需的信息将由节点本身携带。赋值是要求您更改树实现以使用该方法，而不是现在使用的基于堆栈的方法。

拥有“后退”或父指针在某些方面很方便，但在其他方面则不方便。它们为许多事物提供了更简单的表达式，并且在树遍历期间需要更少的簿记。它们还可以让您在树函数之间更有效地共享代码。另一方面，无论何时何地修改树，它们都是一个额外的项目，并且它们会引入冗余，因为它们会产生不一致的机会。

您的作业首先添加一个指向struct tr_n_t的指针。然后，无论何时向树中添加节点，都必须正确地初始化它，并在重新定义节点作为删除的直接结果或重新平衡过程时更新它。您将进一步删除insert()和delete()中跟踪通过树的路径到要删除的插入点/节点的代码，并修改两个函数中的重新平衡代码，以便它使用新的指向要回溯树而不是像现在一样使用堆栈。